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Thread: Suppose we have a right triangle with integer sides

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    Suppose we have a right triangle with integer sides

    Suppose we have a right triangle with integer sides (in some unit of measure). Prove
    (a) One of the legs has a length divisible by 3.
    (b) One of the three sides has length divisible by 5.
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    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by NikoBellic View Post
    Suppose we have a right triangle with integer sides (in some unit of measure). Prove
    (a) One of the legs has a length divisible by 3.
    We want to show either $\displaystyle m^2-n^2 $ or $\displaystyle 2mn $ is divisible by $\displaystyle 3 $.

    WLOG if $\displaystyle m\equiv0\bmod{3} $ then we're done since then $\displaystyle 2mn\equiv0\bmod{3} $.

    So assume $\displaystyle m $ and $\displaystyle n $ are both not divisible by $\displaystyle 3 $.

    But then by flt we get that $\displaystyle m^2\equiv1\bmod{3} $ and $\displaystyle n^2\equiv1\bmod{3} $, so $\displaystyle m^2-n^2\equiv0\bmod{3} $.
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